RELATIONSHIP OF CLINICAL FACTORS WITH ADIPONECTIN AND LEPTIN IN CHILDREN WITH NEWLY DIAGNOSED TYPE 1 DIABETES. Yuan Gu

Transcription

1 RELATIONSHIP OF CLINICAL FACTORS WITH ADIPONECTIN AND LEPTIN IN CHILDREN WITH NEWLY DIAGNOSED TYPE 1 DIABETES by Yuan Gu BE, Nanjing Institute of Technology, China, 2006 ME, University of Shanghai for Science and Technology, China, 2009 Submitted to the Graduate Faculty of the Department of Biostatistics Graduate School of Public Health in partial fulfillment of the requirements for the degree of Master of Science University of Pittsburgh 2015

2 UNIVERSITY OF PITTSBURGH GRADUATE SCHOOL OF PUBLIC HEALTH This thesis was presented by Yuan Gu It was defended on June 17, 2015 and approved by Ingrid M. Libman, MD, PhD Assistant Professor, Pediatrics Children s Hospital, University of Pittsburgh Medical Center Francis Pike, PhD, Assistant Professor, Critical Care Medicine, School of Medicine, University of Pittsburgh Evelyn O. Talbott, DrPH, MPH Professor, Epidemiology, Graduate School of Public Health, University of Pittsburgh Thesis Advisor: Vincent C. Arena, PhD Associate Professor, Biostatistics, Graduate School of Public Health, University of Pittsburgh ii

3 Copyright by Yuan Gu 2015 iii

4 Vincent C. Arena, PhD RELATIONSHIP OF CLINICAL FACTORS WITH ADIPONECTIN AND LEPTIN IN CHILDREN WITH NEWLY DIAGNOSED TYPE 1 DIABETES Yuan Gu, MS University of Pittsburgh, 2015 ABSTRACT Aim: To investigate potential predictors for adiponectin, leptin, and adiponectin/leptin ratio in children with newly diagnosed type 1 diabetes (T1D). Methods: Medical records were reviewed from 175 subjects (165 Caucasian, 8 African American (AA), 59.4% male, mean age 9.7 ± 3.8 yrs) with new onset T1D diabetes diagnosed between January 2004 and December 2006 at Children s Hospital of Pittsburgh of UPMC. Adiponectin, leptin, islet cell autoantibodies including ICA, glutamic acid decarboxylase (GAD) (65 kda isoform), insulin antibody (IA2), insulin autoantibody (IAA) and zinc transporter 8 (ZnT8A), anthropometric and clinical variables including systolic and diastolic blood pressure, height, weight, waist circumference (with calculation of body mass index (BMI), waist percentile and waist/height ratio) and insulin dose, laboratory data including hemoglobin A1c(HbA1c), glucose, lipid profile (low-density lipoprotein (LDL) and high-density lipoprotein (HDL), cholesterol, and triglycerides) and C-peptide were all measured at 3 months after start of insulin therapy. HLA typing was determined for the presence of the DQ2 and/or DQ8 haplotypes. Results: Univariate and multivariate linear regression analyses were performed assessing factors related with adiponectin and leptin, using two different procedures. Nine candidate models were identified and examined for consistency. Adiponectin was significantly associated with age, waist percentile and greater number of positive antibodies. Leptin was significantly associated iv

5 with gender, BMI z-score, central obesity, C-peptide, GAD, HbA1c, and insulin dose adjusted by HbA1c. Adiponectin/leptin ratio was significantly associated with gender, age waist percentile, waist/height ratio, insulin dose adjusted by HbA1c, HbA1c, glucose, and C-peptide. Public health focused conclusion: Adiponectin, leptin and adiponectin/leptin ratio had different significant predictors. However there were a set of factors that where in common. Insulin resistance has been recognized to be present in youth with T1D. Adiponectin and leptin have an influence on insulin sensitivity. Identifying the significant predictors for these hormones may contribute to our understanding of their role in the pathogenesis of T1D. The identification of potential modifiable risk factors in children with this condition would be high priority. v

6 TABLE OF CONTENTS PREFACE... XIV 1.0 INTRODUCTION METHODS DATA PREPARATION INDEPENDENT VARIABLES Demographic variables Age and gender Measure of Adiposity Measure of autoimmunity Clinical and laboratory measures DQ2/DQ Derived variables Categorical groups for number of positive antibodies Insulin dose adjusted by HbA1c Zinc transporter 8 autoantibodies classes ( + and - ) Glucose categorical classes ( <120 and >= 120) STATISTICAL ANALYSIS RESULTS... 9 vi

7 3.1 DEPENDENT VARIABLES SUMMARY INDEPENDENT VARIABLES SUMMARY Continuous independent variables Categorical variables CORRELATIONS UNIVARIATE LINEAR REGRESSION WITH SIMPLE ADJUSTMENT Simple regression of log (adiponectin) Simple regression of log(leptin) Simple regression of log(adiponectin/leptin) Comparison of univariate regression results MULTIVARIATE LINEAR REGRESSION Independent variables selection procedures for multivariate regression models Final models of log (adiponectin) Final models of log (leptin) Final models of log (adiponectin/leptin) Results of multivariate regression models TEST OF REGRESSION MODEL ASSUMPTIONS Existence Independence Linearity Linearity testing for log (adiponectin) vii

8 Linearity testing for log (leptin) Linearity testing for log (adiponectin/leptin) Homoscedasticity Normality Leverage and influence points diagnostics Leverage points Influential points Collinearity DISCUSSION AND CONCLUSION APPENDIX: SAS PROGRAM FOR REGRESSIONS BIBLIOGRAPHY viii

9 LIST OF TABLES Table 1. Statistical summary for dependent variables... 9 Table 2. Kolmogorov-Smirnov test for normality on dependent variables Table 3. Statistical summary for continuous independent variables Table 4. Statistical summary for categorical independent variables Table 5. Pearson correlation coefficients for continuous independent variables Table 6. Pearson correlation coefficients for continuous independent variables Table 7. Highly correlated groups among independent variables Table 8. Univariate regression and univariate regression with age adjustment of log (adiponectin) Table 9. Univariate regression and univariate regression with BMI percentile adjustment of log (adiponectin) Table 10. Univariate regression with C-peptide, glucose of log (adiponectin) Table 11. Univariate regression and univariate regression with age adjustment of log (leptin).. 22 Table 12. Univariate regression and univariate regression with BMI percentile adjustment of log (leptin) Table 13. Simple regression with C-peptide, glucose of log (leptin) ix

10 Table 14. Univariate regression and univariate regression with age adjustment of log (adiponectin/leptin) Table 15. Univariate regression and univariate regression with BMI percentile adjustment of log (adiponectin/leptin) Table 16. Simple regression with C-peptide, glucose of log (adiponectin/leptin) Table 17. Comparison of univariate regression results Table 18. Multivariate linear regression models of log (adiponectin) Table 19. Multivariate linear regression models of log (leptin) Table 20. Multivariate linear regression models of log (adiponectin/leptin) Table 21.VIFs for models 1 to Table 22.VIFs for models 6 to Table 23.VIFs for model 8 to x

11 LIST OF FIGURES Figure 1. Comparison histograms of adiponectin, leptin, adiponectin/leptin ratio, Figure 2.Partial regression residual plot for model Figure 3.Partial regression residual plot for model Figure 4. Partial regression residual plot for model Figure 5. Partial regression residual plot for model Figure 6. Partial regression residual plot for model Figure 7. Partial regression residual plot for model Figure 8. Partial regression residual plot for model Figure 9. Partial regression residual plot for model Figure 10. Partial regression residual plot for model Figure 11. Model 1: Scatter plot of standardized residuals versus fitted points Figure 12. Model 1: Scatter plot of outcome variables versus fitted points Figure 13. Model 2: Scatter plot of standardized residuals versus fitted points Figure 14. Model 2: Scatter plot of outcome variables versus fitted points Figure 15.Model 3: Scatter plot of standardized residuals versus fitted points Figure 16. Model 3: Scatter plot of outcome variables versus fitted points Figure 17.Model 4: Scatter plot of standardized residuals versus fitted points Figure 18.Model 4: The scatter plot of outcome variables versus fitted points xi

12 Figure 19.Model 5: Scatter plot of standardized residuals versus fitted points Figure 20. Model 5: Scatter plot of outcome variables versus fitted points Figure 21.Model 6: Scatter plot of standardized residuals versus fitted points Figure 22.Model 6: Scatter plot of outcome variables versus fitted points Figure 23.Model 7: Scatter plot of standardized residuals versus fitted points Figure 24.Model 7: Scatter plot of outcome variables versus fitted points Figure 25.Model 8: Scatter plot of standardized residuals versus fitted points Figure 26. Model 8: Scatter plot of outcome variables versus fitted points Figure 27.Model 9: Scatter plot of standardized residuals versus fitted points Figure 28.Model 9: Scatter plot of outcome variables versus fitted points Figure 29.Histogram and QQ plot of standardized residual versus normal distributions for model Figure 30. Histogram and QQ plot of standardized residual versus normal distributions for model Figure 31. Histogram and QQ plot of standardized residual versus normal distributions for model Figure 32.Histogram and QQ plot of standardized residual versus normal distributions for model Figure 33.Histogram and QQ plot of standardized residual versus normal distributions for model Figure 34. The histogram and QQ plot of standardized residual versus normal distributions based on model xii

13 Figure 35.Histogram and QQ plot of standardized residual versus normal distributions for model Figure 36.Histogram and QQ plot of standardized residual versus normal distributions for model Figure 37.Histogram and QQ plot of standardized residual versus normal distributions for model Figure 38. Leverage versus outcome variables for models 1to Figure 39.Cook s Distance versus outcome variables for models 1 to xiii

14 PREFACE Thanks very much for all the help from my respected academic advisor, Dr. Vincent Arena. All analysis was under his careful guidance and advice. Thank you for all of your patience and for helping me understood a most important concept: How to do a project systematically. To Drs. Natalie Hecht Baldauff and Ingrid Libman, thank you for your support and providing me with clinical explanations, interpretations and corrections, as well as your kind patience during the project. To the rest of my committee Drs. Evelyn Talbott and Francis Pike, thank you for your constructive suggestions. To dear Joanne Pegher, thank you for all your help editing my paper and your patience. To Ms. Renee Valenti, thank you for your support during my time at GSPH. Acknowledgement is given to Dr. Dorothy Becker for the use of data from the Juvenile Onset Diabetes Project. This work was supported by National Institutes of Health (NIH) grants R01 DK46864 (D. Becker), Grant Numbers UL1 RR and UL1TR (PCTRC). xiv

15 1.0 INTRODUCTION Type 1diabetes (T1D) is the most common form of diabetes mellitus in children and young adults. Although there has been substantial progress in the knowledge of the pathogenesis and natural history of T1D in recent years, there is no effective treatment available to cure the disease [1,2].Worldwide, the incidence of T1D continues to increase at a rate of nearly 3% per year [3]. In 2011, an estimated 490,100 children, worldwide, below the age of 15 years were living with T1D [4]. The obesity epidemic is widely blamed for a startling rise in the incidence of type 2 diabetes (T2D) among children. Intriguing new research suggests it may also play a role for the increase in T1D [5]. A marked increase in the prevalence of overweight and obesity in type T1D has been demonstrated [6]. Insulin resistance may play a role in the pathogenesis of T1D [7]. Adiponectin and leptin are hormones secreted by the adipose tissue and have an influence on insulin sensitivity [8]. Adiponectin levels are low in human obesity, cardiovascular disease, and T2D. Paradoxically, high adiponectin levels, specifically the high molecular weight isoform, have been reported in established T1D. Leptin, the adipocytokine product of the Ob(Lep) gene, reflects the degree of adiposity and is stimulated by insulin, rising acutely with insulin therapy in both in vitro rodent studies and in children with new onset T1D. The adiponectin/leptin ratio also has been used in studies of T1D and T2D as a surrogate measure of insulin sensitivity [9]. 1

16 Previous studies have explored racial differences in adiponectin and leptin and their relationship with islet autoimmunity in equally matched children of African American (AA) and Caucasian with new onset T1D. Adiponectin levels increase as early as three days after initiation of insulin therapy with a statistically significant increase by day 5. No significant racial differences in adiponectin, leptin, and adiponectin/leptin ratio levels were found after adjustment for BMI. Subjects with higher number of positive autoantibodies had higher adiponectin levels, lower leptin levels, and higher adiponectin/leptin ratios than those with lower numbers of positive antibodies [9]. This study aimed to evaluate adiponectin and leptin measured 3 months after diagnosis of T1D and their relationships with number of islet-cell autoantibodies and measures of adiposity. 2

17 2.0 METHODS 2.1 DATA PREPARATION Two data sets were combined to form the data file used for this analysis. Subjects included were 351 children diagnosed with T1D from January 2004 to December 2006 at the Children s Hospital of Pittsburgh of UPMC. Inclusion criteria was as follows: 1) informed consent signed, 2) diagnosis of diabetes requiring insulin, 3) insulin treatment at time of hospital discharge, and 4) available research laboratory results for three or more β-cell autoantibodies, including islet cell autoantibodies (ICA), glutamic acid decarboxylase (GAD) (65 kda isoform), insulin antibody (IA2), and insulin autoantibody (IAA). Cases with clinical maturity-onset diabetes of the young, T2D, and without AA were excluded. Of the 351 subjects recruited, 295 met the inclusion criteria of 3 or more AA measured [10]. Of these 295 patients, 175 patients had measurements of adiponectin and leptin and were included in the analysis. 3

18 2.2 INDEPENDENT VARIABLES Demographic variables Age and gender Age and gender were considered as potential predictors or confounders of adiponectin, leptin and adiponectin/leptin ratio Measure of Adiposity Body Mass Index (BMI), BMI percentile (BMI %), BMI z-score, height, weight BMI was defined as weight/height 2 (kg/m 2 ). Height (cm) and weight (kg) were collected to calculate BMI. Instead of using fixed BMI values to classify individuals (as typically done with adults), children s BMI was classified using thresholds that vary to take into account the child s age and gender. BMI thresholds were defined in terms of a specific BMI z-score, or BMI percentile (BMI %), on a child growth reference [11]. Waist, Waist/height Ratio, Central Obesity The waist circumference (cm) and waist/height ratio were two indicators of obesity among young children [12]. Waist/height ratio had been proposed as an easily measurable anthropometric index for detection of central obesity [13]. Central obesity was defined as having a ratio exceeding 0.5. Waist circumference was the actual recording of circumference in centimeters. In addition, waist percentile (waist %) was available and is standardized to age and sex specific norms for children. 4

19 2.2.2 Measure of autoimmunity Initially, four types of antibodies including insulin antoantibodies(iaa,) insulin antibody(ia2), islet cell antibodies(ica), and glutamic acid decarboxylase(gad) (65 kda isoform) were measured. The appearance of autoantibodies to one or several of the autoantigens signaled an underlying autoimmune process [14]. Each antibody was classified as having a positive or negative response in the analysis. Additionally, a composite variable for the number of positive antibodies among IAA, ICA, IA2 and GAD was created. In addition, information was available for a fifth autoantibody, Zinc transporter 8(ZnT8A) Studies following the T1D progression from the prodrome stage to onset among high risk patients found that the emergence of ZnT8A autoantibody usually preceded the appearance of T1D clinical symptoms and persisted to disease onset [15] Clinical and laboratory measures Blood pressure was assessed and percentiles were determined based on gender, age and height. Laboratory measures included HbA1c and glucose. HbA1c measured glycated hemoglobin and indicated the average blood sugar levels in the past two to three months. Lipids including LDL, HDL, cholesterol and triglycerides, C-peptide and insulin dose were measured. 5

20 2.2.4 DQ2/DQ8 HLA typing was available for the presence of DQ2, DQ8, DQ2 /DQ8. These haplotypes are related to risk of diabetes and were also considered as potential confounders of adiponectin, leptin and the adiponectin/leptin ratio in this study Derived variables Some new independent variables were created for investigating more specific statistical relationship between predictors and outcomes Categorical groups for number of positive antibodies The values of number of positive antibodies ranged from 0 to 4.We examined the number of positive antibodies by using several different groupings. Class 1: 0 versus people who have 0 positive antibody; people who have more than 1 positive antibodies. Class 2: 0-1 versus people who have 0 or 1 positive antibody; people who have 2, 3 or 4 positive antibodies. Class 3: 0, 1, 2 versus people who have 0 positive antibody; 1--- people who have 1 positive antibody; people who have 2 positive antibodies; 6

21 3 --- people who have 3 or 4 positive antibodies Insulin dose adjusted by HbA1c Insulin dose adjusted by HbA1c =HbA1c (percentile) + [4 * insulin dose (units per kilogram per 24 h)] In previous studies, a negative association between stimulated C-peptide and HbA1c (regression coefficient -0.21, P < 0.001) and insulin dose (-0.94, P < 0.001) was shown. Insulin dose adjusted by HbA1c reflected residual β-cell function and had better stability compared with the conventional definitions [16] Zinc transporter 8 autoantibodies classes ( + and - ) The ZnT8A variable was developed into two categories. When ZnT8A was greater than 0.02, then the class was defined as +, when ZnT8A was less than 0.02, the class was defined as Glucose categorical classes ( <120 and >= 120) The glucose variable was dichotomized into two categories: greater than or equal to 120 and less than

22 2.3 STATISTICAL ANALYSIS All continuous variables of interest were described by means, standard deviations, number of missing, number of observations, medians and percentages. For all the categorical variables, proportion summaries were used. Log transformations of adiponectin, leptin and adiponectin/leptin ratio were created to normalize the distributions of the outcome variables. The log scale of the three outcome variables would be used as the dependent variables in the regression analysis. Correlations between different independent continuous variables were assessed using Pearson correlation coefficients. Univariate linear regression and multivariate linear regression analysis were performed to investigate potential predictors of adiponectin, leptin and adiponectin/leptin ratio separately. Analyses were performed using SAS 9.3. All statistical hypothesis testing was conducted as two-sided tests, and with statistical significance p <

23 3.0 RESULTS 3.1 DEPENDENT VARIABLES SUMMARY Table 1. Statistical summary for dependent variables Variable Minimum Maximum Mean Median N N Std Lower Upper Skewness miss Dev Quartile Quartile Adiponectin(ug/ml) Leptin(ug/ml) Adiponectin/leptin ratio Log(adiponectin) Log(leptin) Log (adiponectin/leptin) As shown in Table 1, 15 observations were missing for leptin and adiponectin/leptin ratio. The range of adiponectin was from 2 to 54ug/ml, the range of leptin was from 0.78 to 58.6ug/ml, and the ratio was from 0.11 to All distributions of the three variables were right skewed (Figure 1). Using the log transformation, the skewness was reduced near to 0. Using the Kolmogorov-Smirnov test for normality (Table 2), all three transformed variables were normally distributed, with each p-value >0.15. In the regression analysis, the log transformed variables would be used as the outcome variables. 9

24 Figure 1. Comparison histograms of adiponectin, leptin, adiponectin/leptin ratio, log (adiponectin),log (leptin), log (adiponectin/leptin) and normal distribution 10

25 Table 2. Kolmogorov-Smirnov test for normality on dependent variables Variable P-value for Kolmogorov-Smirnov test Log(adiponectin) >0.150 Log(leptin) >0.150 log (adiponectin/leptin) > INDEPENDENT VARIABLES SUMMARY Continuous independent variables Table 3. Statistical summary for continuous independent variables Variable Minimum Maximum Mean Median N N Miss Std Dev Lower Quartile Upper Quartile BMI % BMI z-score Waist/height Age(months) Height(cm) IAA ICA GAD IA HbA1c Cholesterol LDL HDL Triglyceride Diastolic blood pressure 11

26 Table 3 Continued percentile Systolic blood pressure percentile Glucose C-peptide Insulin dose ZnT8A Descriptive results for the continuous independent variables were provided in Table 3. The mean level of BMI percentile was 74% and the lower quartile was 60.3%, which indicated that the subjects in this study were heavier than typical child at their same age. Ages of the patients ranged from 17 to 230 months (1.4 to 19.2 yrs). The mean age was months (9.7 yrs), close to the median age, which was 117 months (9.8 yrs). Waist/height ratios ranged from 0.24 to 0.67.The mean value was 0.47, close to the lower quartile value of 0.43, and the upper quartile value was The waist/height ratios of most observations were between 0.43 and 0.51, and did not indicate central obesity (waist/height ratio 0.5). Values for the other variable values were all in expected ranges. There was an appreciable number of subjects missing IAA (n=125) and LDL (n=78) information which limited our ability to include these variables in further analysis. In the later analysis, the two variables would be excluded Categorical variables For the categorical variables, the frequency and missing numbers were provided by PROC FREQ in SAS 9.3 (Table 4). Over half were males (59.4%) and 42 (29.4%) had central 12

27 obesity participated in the study. Over 90% of the subjects had at least one positive antibody. With respect to DQ2/DQ8 less than 20% of the subjects were in the low risk group of X. For waist percentile, only 15.9% of the subjects were considered having large waist circumferences. Table 4. Statistical summary for categorical independent variables Variables Values Frequency Percent Frequency Missing Gender Central obesity DQ2/DQ8 Waist % Number of positive antibodies Male Female No Yes X DQ DQ DQ2/DQ <25% %-75% >75% CORRELATIONS The Pearson correlation coefficients assessed the linear associations between different continuous variables. The correlation coefficient between log (adiponectin) and log (leptin) was - 13

28 0.0096, with a p-value of 0.9, which indicated that log (adiponectin) and log (leptin) were not linearly associated. In the correlation matrix (Table 5 and Table 6), the red shaded values indicated correlations greater than 0.75 and the blue shade values were from 0.5 to The two colors indicated that the variables were highly correlated. The cyan shaded values indicated moderate correlations from 0.25 to 0.5. Careful consideration as to which variables of the highly correlated factors should be used in the regression models. Groups of highly correlated variables were summarized in Table 7. Table 5. Pearson correlation coefficients for continuous independent variables BMI % BMI Waist/heig Heig GA HbA Cholester Waist Age IAA ICA IA2 z-score ht ht D 1c ol BMI %* <.0001 <.0001 < BMI z-score <.0001 <.0001 < Waist <.0001 <.0001 <.0001 <.0001 < Waist/height <.0001 <.0001 < Age < < < Height < < < IAA ICA < GAD IA < HbA1c <.0001 < Cholesterol LDL <.0001 HDL <.0001 <.0001 <.0001 < <.0001 Triglyceride

29 Table 5 Continued Percentile for diastolic blood pressure Percentile for systolic blood pressure <.0001 < <.0001 <.0001 < Glucose Insulin dose adjusted by HbA1c < <0.01 < < < C-peptide <.0001 < < Insulin Dose < ZnT8A < < *The first line for each variables was the correlation coefficient, the second line is the p-value (Prob > r under H0: Rho=0). Table 6. Pearson correlation coefficients for continuous independent variables LDL HDL Triglyceride Percentile for diastolic blood presuure Percentile for systolic blood presuure 15 Glucose Insulin dose adjusted by HbA1c C- peptide Insulin dose BMI %* BMI z-score < Waist < < Waist/height < < Age <.0001 <.0001 < < Height <.0001 <.0001 < < IAA ICA <.0001 GAD IA <.0001 HbA1c <.0001 <.0001 <.0001 < Cholesterol <.0001 < LDL HDL ZnT8A

30 Table 6 Continued 0.93 < Triglyceride Percentile for diastolic blood pressure Percentile for systolic blood pressure 0.82 < < < Glucose Insulin adjusted HbA1c dose by <.0001 < C-peptide < Insulin dose < ZnT8A *The first line for each variables was the correlation coefficient, the second line is the p-value (Prob > r under H0: Rho=0). Table 7. Highly correlated groups among independent variables Highly correlated groups Variables Pearson Correlation Coefficients BMI % & BMI z-score : BMI %, BMI z-score, waist/height, waist BMI % & waist/height 0.63 BMI z-score & waist 0.58 BMI z-score & waist/height 0.73 Waist & waist/height : Waist, waist/height, age, height Waist & age 0.68 Age & height 0.95 Height & waist : LDL, cholesterol LDL & cholesterol : Insulin dose adjusted by HbA1c, Insulin dose, HbA1c Insulin dose adjusted by HbA1c & HbA1c Insulin dose adjusted by HbA1c & insulin dose

31 3.4 UNIVARIATE LINEAR REGRESSION WITH SIMPLE ADJUSTMENT Simple regression of log (adiponectin) Table 8. Univariate regression and univariate regression with age adjustment of log (adiponectin) Independent Var β P-value Adjuste d Var β P-value R 2 Pr > F N of missing BMI %* BMI % Age BMI z-score BMI z-score Age Central obesity Central obesity Age Waist Waist Age Height Height Age Waist/height Waist/height Age HbA1c HbA1c Age Cholesterol Cholesterol Age LDL LDL Age HDL HDL Age DQ2/DQ X 0 0 DQ DQ2/DQ8 class DQ DQ2/DQ Age X 0 0 DQ

32 Table 8 Continued DQ2/DQ8 class DQ Triglycerides Triglycerides Age Percentile for diastolic blood pressure Percentile for diastolic blood Age pressure Percentile for systolic blood pressure Percentile for systolic blood Age pressure *The shaded rows are univariate regression analysis without adjustment for age. Table 8 provided the results of the univariate regression and univariate regression with age adjustment of log (adiponectin). P-value: The p-values indicated if the independent variable was significant for predicting the outcome variable. In the univariate regression analysis, only waist, height, and waist/height ratio were significant for predicting log (adiponectin). After adjusting for age, waist and height became non-significant. However, waist/height ratio was still significant. This change indicated that after accounting for the effect of age on adiponectin, waist and height no longer added to the model. β coefficients: The β coefficients of waist, height and waist/height ratio in univariate regression models were negative, indicating a negative linear relationship between the three independent variables and log (adiponectin). R 2 : Although in the univariate analysis, waist, height and waist/height ratio were significant, the R 2 were very small, demonstrating that the variation explained by those variables were very low, only between 4 to 8 percent. Even after age adjustment, the amount of variation explained remained small. 18

33 All β coefficients were small, close to zero. This was in part because the univariate regression models only predicted small percentages of the variability of the dependent variable. Covariate adjustment variable: The p-values indicated that age was significant for predicting log (adiponectin). The β coefficients indicated that age had a negative linear relationship with log (adiponectin). Table 9. Univariate regression and univariate regression with BMI percentile adjustment of log (adiponectin) Independent Var β P-value Adjusted Var β P-value R 2 Pr > F N of missing IA IA BMI % IAA IAA BMI % ICA ICA BMI % GAD GAD BMI % ZnT8A ZnT8A BMI % Zinc category (+,-) Zinc category (+,-) BMI % Insulin dose Insulin dose BMI % Number of positive antibodies Number of positive antibodies BMI %

34 Table 9 Continued Number of positive antibodies (continuous) Number of positive antibodies BMI % (continuous) Number of positive antibodies class1(0, 1) Number of positive antibodies class1(0, BMI % ) Number of positive antibodies class2(0-1, 2) Number of positive antibodies class2(0-1, 2) Number of positive antibodies class3(0,1,2, 3) Number of positive antibodies BMI % class3(0,1,2, 3) Table 9 provided the results for the univariate regression and univariate regression with BMI percentile adjustment of log (adiponectin). 20

35 P-value: In the univariate regression analysis, number of positive antibodies (continuous), number of positive antibodies class2 and number of positive antibodies class3 were significant for predicting log (adiponectin) with p-values of 0.042, and With the addition of BMI percentile to the model, the p-values for number of positive antibodies (continuous), number of positive antibodies class2 and class3 increased to 0.054, and β coefficients : The β coefficients for number of positive antibodies (continuous), number of positive antibodies class2 and class3 indicated the negative linear relationship with log(adiponectin) before and after BMI percentile adjustment. R 2 : In the univariate analysis of number of positive antibodies (continuous), number of positive antibodies class2 and class3, the R 2 values were very small, revealing the variation explained from those variables were very low, only from 2 to 5 percent. With the addition of BMI percentile to the models, the R 2 were still very small. Covariate adjustment variable: BMI percentile was not significant for predicting the log (adiponectin). Table 10. Univariate regression with C-peptide, glucose of log (adiponectin) Independent Var1 β P-value Independent Var2 β P-value R 2 Pr > F N of missing Glucose C-peptide C-peptide/glucose C-peptide Glucose C-peptide Glucose(<120, >=120) Table 10 provided the results for the univariate regression analysis when glucose and C- peptide were treated as predictors of log (adiponectin). P-value: None of independent variables was significant. 21

36 3.4.2 Simple regression of log(leptin) Table 11. Univariate regression and univariate regression with age adjustment of log (leptin) Independent Var β P-value Adjusted Var β P-value R 2 Pr > F N of missing BMI % < BMI % <.0001 Age < BMI z-score 0.4 < BMI z-score 0.4 <.0001 Age < Central obesity 0.64 < Central obesity 0.63 <.0001 Age < Waist < Waist Age < Height Height Age Waist/height Waist/height Age < HbA1c HbA1c Age Cholesterol Cholesterol Age LDL LDL Age HDL HDL Age DQ2/DQ X 0 0 DQ DQ2/DQ8 class DQ DQ2/DQ Age X 0 0 DQ DQ2/DQ8 class DQ Triglycerides Triglycerides Age Percentile for diastolic

37 Table 11 Continued blood pressure Percentile for diastolic Age blood pressure Percentile for systolic blood pressure Percentile for systolic Age blood pressure Table 11 provided the results for the univariate regression and univariate regression with age adjustment of log (leptin). P-value: In the univariate regression analysis, BMI percentile, BMI z-score, central obesity waist, height and waist/height ratios were significant for predicting log (leptin). When the covariate of age was added to the models, those variables were still significant except height, due to the high correlation between age and height. β coefficients: The β coefficients of BMI percentile, BMI z-score, central obesity, waist, height and waist/height ratios in univariate regression models were all positive, revealing the positive linear relationship with log (leptin). R 2 : The R 2 of the univariate regression models of BMI percentile, BMI z-score, central obesity, waist, height and waist/height ratios were very small, demonstrating that the variation explanation from those variables were very low, no more than 5 percent. After age was adjusted, the effect of predicting variation increased by nearly 10 percent, and ranged from 15 to 18 percent. Covariate adjustment variable: Age was significant for predicting log (leptin). However, in the model of waist and height, age reduced its significant importance because of high correlations between age and height and age and waist. Age had positive linear relationship with log (leptin), indicating that when age increased, log (leptin) also increased. 23

38 Other variables: When age was added as a covariate, it had an influence on the relationships between HbA1c and log (leptin), percentile for diastolic blood pressure and log (leptin) and percentile for systolic blood pressure and log (leptin). Although p-values became smaller, none reached statistical significance. Table 12. Univariate regression and univariate regression with BMI percentile adjustment of log (leptin) Independent Var β P-value Adjusted Var β P-value R 2 Pr > F N of missing IA IA BMI % IAA IAA BMI % ICA ICA BMI % < GAD GAD BMI % < < ZnT8A ZnT8A BMI % Zinc category (+,-) Zinc category (+,-) BMI % < Insulin dose Insulin dose BMI % < Number of positive antibodies Number of positive antibodies BMI % Number of positive

39 Table 12 Continued antibodies (continuous) Number of positive antibodies (continuous) Number of positive antibodies class1(0, 1) Number of positive BMI % < antibodies class1(0, 1) BMI % < Number of positive antibodies class2(0-1, ) Number of positive antibodies class2(0-1, < ) Number of positive antibodies class3(0,1,2, ) Number of positive antibodies class3(0,1,2, BMI % ) Table 12 provided the statistical information about the univariate regression and univariate regression with BMI percentile adjustment of log (leptin). P-value: In the univariate regression analysis, ZnT8A and insulin dose were significant for predicting log (leptin). When age was added into the model, insulin dose was still significant but ZnT8A became non-significant. 25

40 β coefficients: The β coefficient for insulin dose revealed positive linear relationship with log (leptin) when age was adjusted or not. R 2 : The R 2 was very small, demonstrating that the variation explained from insulin dose was very low(9.9%). When BMI percentile included in the model, the R 2 increased to 17.6%. Covariate adjustment variable: BMI percentile was significant for predicting log (leptin) and had positive linear relationship with log (leptin). Table 13. Simple regression with C-peptide, glucose of log (leptin) Independent Var1 β P-value Independent Var2 β P-value R2 Pr > F N of missing Glucose C-peptide C-peptide/ glucose C-peptide Glucose C-peptide Glucose(<120, >=120) Table 13 provided the results of simple regression with C-peptide and glucose of log (leptin). P-value: All the predictors of glucose and C-peptide were significant for predicting log (leptin). β coefficients: The β coefficient of glucose showed a negative linear relationship between glucose and log (leptin). The β coefficients of other variables were all positive, revealing the positive linear relationship between the other variables and log (leptin). R 2 : The R 2 were very small, from 2% to 9%. 26

41 3.4.3 Simple regression of log(adiponectin/leptin) Table 14. Univariate regression and univariate regression with age adjustment of log (adiponectin/leptin) Independent Var Β P-value Adjusted Var β P-value R 2 Pr > F N of missing BMI % BMI % Age < < BMI z-score < BMI z-score <.0001 Age < < Central obesity Central obesity Age < < Waist < Waist Age < Height < Height Age < Waist/height Waist/height Age < < HbA1c E HbA1c Age < < Cholesterol Cholesterol Age < < LDL LDL Age < < HDL HDL Age < < DQ2/DQ X 0 0 DQ DQ2/DQ8 class DQ DQ2/DQ Age < X 0 0 DQ DQ2/DQ8 class DQ Triglycerides

42 Table 14 Continued Triglycerides 7.5E Age < < Percentile for diastolic blood pressure Percentile for diastolic blood pressure Percentile for systolic blood pressure Percentile for systolic blood pressure Age Age Table 14 provided the results of the univariate regression and univariate regression with age adjustment of log (adiponectin/leptin). P-value: In the univariate regression, BMI percentile, BMI z-score, central obesity, waist, height and waist/height ratios were significant for predicting log (adiponectin/leptin). When age was added into the model, those variables were still significant except height, due to the high correlation between age and height. β coefficients: The β coefficients of BMI percentile, BMI z-score, central obesity, waist, height and waist/height ratios in univariate regression models were negative, revealing a negative linear relationship between those variables and log (adiponectin/leptin). R 2 : In the univariate analysis, the R 2 indicated that the variation explained from those variables were from 6% to 20%. After the addition of age into the model, the percent of variation explained in the dependent variable increased from 20% to 23%. Covariate adjustment variable: Age was significant for predicting log (adiponectin/leptin). However, with the addition of waist and height, age became non-significant because of high correlations between age and height and age and waist. Age had negative linear relationship with log (adiponectin/leptin), indicating that when age increased, log (adiponectin/leptin) decreased. 28

43 Table 15. Univariate regression and univariate regression with BMI percentile adjustment of log (adiponectin/leptin) Independent Var β P-value Adjusted Var β P-value R 2 Pr > F N of missing IA IA BMI % IAA IAA BMI % ICA ICA BMI % GAD GAD BMI % ZnT8A ZnT8A BMI % Zinc category (+,-) Zinc category (+,-) BMI % Insulin dose Insulin dose BMI % < Number of positive antibodies Number of positive antibodies BMI % Number of positive antibodies (continuous) Number of positive antibodies (continuous) BMI % Number of positive

44 Table 15 Continued antibodies class1(0, 1) Number of positive antibodies class1(0, 1) BMI % Number of positive antibodies class2(0-1, ) Number of positive antibodies class2(0-1, BMI % ) Number of positive antibodies class3(0,1,2, ) Number of positive antibodies class3(0,1,2, BMI % ) Table 15 provided the results of the univariate regression and univariate regression with BMI percentile adjustment of log (adiponectin/leptin). P-value: In the univariate regression analysis, only insulin dose was significant for predicting log (adiponectin/leptin). After age was adjusted, insulin dose was still significant. β coefficients: The β coefficient of insulin dose in univariate regression model was negative, revealing the negative linear relationship between insulin dose and log (adiponectin/leptin). 30

45 R 2 : The R 2 was small, demonstrating that the variation explanation from insulin dose was 7.5%. When BMI percentile was added into the model, the effect of predicting variation of insulin dose increased to 12.8%. Covariate adjustment variable: BMI percentile was significant for predicting log (adiponectin/leptin) and had a negative linear relationship with log (adiponectin/leptin). Table 16. Simple regression with C-peptide, glucose of log (adiponectin/leptin) Independent Var1 β p-value Independent Var2 β p-value R2 Pr > F N of missing Glucose C-peptide C-peptide/ glucose C-peptide Glucose C-peptide Glucose(<120, >=120) Table 16 provided the results of simple regression with C-peptide and glucose of log (adiponectin/leptin). P-value: All of the glucose and C-peptide variables were significant predictors of log (adiponectin/leptin). β coefficients: Glucose had a positive β coefficient, revealing the positive linear relationship between glucose and log (adiponectin/leptin). The β coefficients of the other predictors were all negative, revealing a negative relationship with log (adiponectin/leptin). R 2 : The R 2 were very small, from 2% to 10% Comparison of univariate regression results From Table 17, summarized the result of the univariate regression models 31

46 Number of positive antibodies classes: For log (leptin) and log (adiponectin/leptin), the number of positive antibody level was not significant as a predictor. When the outcome variable was log (adiponectin), the number of positive antibody level was significant. The β coefficient was negative, which revealed when the number of positive antibodies level increased, log (adiponectin) decreased. Waist, height and waist/height ratio: All the three variables were significant predictors for the three outcomes. For log (leptin), waist, height and waist/height had positive β coefficients. For log (adiponectin) and log (adiponectin/leptin), waist, height and waist/height had negative β coefficients. The remaining variables: The remaining variables were significant only for log (leptin) and log (adiponectin/leptin) and their associations were in opposite directions as indicated by sign of the β coefficients. Table 17. Comparison of univariate regression results Significant predictor β coefficient sign for log(adiponectin) β coefficient sign for log(leptin) β coefficient sign for log(adiponectin/leptin) Waist Height Waist/height Number of positive antibodies (continuous) Not significant Not significant Number of positive antibodies class2(0-1, 2) Not significant Not significant Number of positive antibodies class3(0,1,2, 3) Not significant Not significant BMI % Not significant BMI z score Not significant Central obesity Not significant ZnT8A Not significant Not significant Insulin dose Not significant Glucose Not significant C-peptide Not significant C-peptide/ glucose Not significant 32

47 3.5 MULTIVARIATE LINEAR REGRESSION Independent variables selection procedures for multivariate regression models To select the independent variables of the multivariate regression models, two selection procedures were performed in parallel and compared: manual, and a combination automatic + manual selection procedure. All multivariate regression models were fit using PROC GLM in SAS 9.3. Manual selection procedure: An initial model was run including all non-correlated variables and one variable from each of the five highly correlated groups (Table 7). Each of the variables from the five highly correlated groups was chosen randomly for inclusion in the initial model. This approach was designed to decrease collinearity resulting from inclusion of more than one variable from a given highly correlated group. For example, in the highly correlated group 1, BMI percentile was used as a first candidate and the other variables in group 1 were left out. All other non-correlated variables which did not belong to the highly correlated groups were added into the initial model simultaneously. Individual predictors in the initial model were considered for exclusion if p-value >0.05, except for BMI percentile in the log(adiponectin) model as this specific relationship was of interest. The individual variable associated with the highest p-value was then excluded, and the model with remaining variables was refit in the next iteration, and the process was repeated as long as the removal of a specific variable did not reverse the significance of variables remaining in the model. For example, if one predictor was excluded with other previously significant variables changed to be nonsignificant, then the variable should be kept in the model. 33

48 If some of the remaining variables in the fitted model were chosen from highly correlated groups, other variables in the same groups were substituted into the model and the results were compared. The variable was kept in the model if it had lowest individual p-value compared to the other variables in the same highly correlated group. The final model was that which included any variables significant at the 0.05 level following this selection procedure, as well as the most significant highly correlated variable within a group. For example, if LDL was tested to be significant, cholesterol would also have been tested in a separate model with the other previously selected variables (based on the p<0.05 criterion). If the model including cholesterol had lower p-value compared to the model including LDL (cholesterol and LDL were in the same highly correlated variable group), then cholesterol would be selected instead of LDL in the final model. Automatic + Manual selection procedure: For this alternative variable selection approach, the independent variables of the initial model included all non-correlated variables and one variable from each of the five highly correlated groups, similar to the manual selection procedure. However, a preliminary round of variable selection was performed by SAS automatically using a stepwise selection criteria (slentry=0.25 slstay=0.15 in SAS code). Once the procedure was completed, the selection was modified manually. For example, if some of the selected predictors were chosen from correlated groups, other variables in the same groups were later substituted for them and the model was refit, similar to the process described above. And using the same manual selection procedure, the variable in the highly correlated group was kept in the model if it had lowest individual p-value. In addition, if the automatic procedure removed BMI percentile in the log(adiponectin) model, then it would be added back (as this specific relationship between BMI percentile and 34

49 log(adiponectin) was of interest). The resulting model after applying this selection protocol was deemed the final model Final models of log (adiponectin) Table 18. Multivariate linear regression models of log (adiponectin) Model Predictors β coefficient DF p- value(overall/in dividual) N of predi ctors N of observati ons Root MSE R 2 Adjust ed R2 AIC BIC BMI % Age <25% -* Waist % 25%-75% >75% Gender Number of positive antibodies class2(0-1, 2) DQ2/DQ8 X - DQ DQ2/DQ DQ BMI % Age <25% - Waist % 25%-75% >75% BMI % Number of positive antibodies Class3(0,1,2, 3) DQ2/DQ8 X - DQ

50 Table 18 Continued DQ2/DQ DQ BMI % Age <25% - Waist % 25%-75% >75% Number of positive antibodies Gender X - DQ2/DQ8 DQ DQ2/DQ DQ BMI % Age <25% - 4 Waist % 25%-75% >75% 0.45 Gender Number of positive antibodies (continuous) DQ2/DQ8 X - DQ DQ2/DQ DQ Waist % <.0001 age <25% %-75%

51 Table 18 Continued >75% Number of positive antibodies class2(0-1, 2) *Reference group Five final models of log (adiponectin) were presented in Table 18. Models 1 to 4 indicated the consistent result with the previous univariate regression result. Model 1: P-value: The results of univariate regression (Table 8) indicated number of positive antibodies class2 had the p-value of After adding BMI percentile, the p-value was , close to the significant level of And the results of multivariate regression model 1showed the p-value for number of positive antibodies class2 was when the other variables (including BMI percentile) were included. Model 2: P-value: The results of univariate regression indicated number of positive antibodies class3 had the p-value of 0.057, close to the significant level of After BMI percentile was included in the model, the p-value was And the results of multivariate regression model 2 showed the p-value of number of positive antibodies class3 was when the other variables (including BMI percentile) were included. Model 3: P-value: When number of positive antibodies was treated as a categorical variable and when BMI percentile was adjusted, in both the multivariate and univariate regression models, number of positive antibodies was not significant for predicting log (adiponectin). 37

52 Model 4: P-value: In the univariate regression, when number of positive antibodies was treated as a continuous variable, it was a significant predictor of log (adiponectin), with a p-value of The p-value increased to after BMI percentile was adjusted. And in the multivariate regression model 4, it was also significant with a p-value of when other variables (including BMI percentile) were included. Models 1 to 4: β coefficients : The β coefficients for the number of positive antibodies classes was negative, indicating that subjects with a greater number of positive antibodies had higher log (adiponectin) level. Waist percentile and age were also significant for predicting log (adiponectin). There was a negative linear relationship between age and log (adiponectin). Individuals in the highest waist percentile group (>75%) had lower log (adiponectin) levels and those in the 25-75% group had higher log (adiponectin) levels compared to individuals in the lowest waist group(<25%). Model 5: Mean squared error root (MSE root), R 2 Adjusted R 2, Akaike information criterion (AIC) and Bayesian information criterion (BIC) of the five models were also presented in Table 18. The five models had similar MSE root, R 2, adjusted R 2, AIC and BIC values. Model 5 was selected by the stepwise criteria in SAS and had the smallest AIC and BIC and did not include BMI percentile. The other four models helped to reveal a relationship of interest between number of positive antibodies and log (adiponectin) when the other variables (including BMI percentile) were included. 38

53 3.5.3 Final models of log (leptin) Table 19. Multivariate linear regression models of log (leptin) Model Predictors Beta coefficient DF p- value(overall/in dividual) N of predi ctors N of observati ons Root MSE R2 Adju sted R2 AIC BIC <.0001 BMI z-score <.0001 C-peptide Age Gender < X - 6 DQ2/DQ8 DQ DQ2/DQ DQ Insulin dose HbA1c Glucose ICA GAD <.0001 BMI z-score C-peptide Central obesity Gender <.0001 Glucose Insulin dose adjusted by HbA1c Two models were presented in Table 19. The final two models of log (leptin) revealed that number of positive antibody was not significant for predicting log (leptin). In model 6, BMI z-score, C-peptide, gender, HbA1c, and GAD were significant. All these variables had a positive linear relationship with log (leptin). In model 7, BMI z-score, C- 39

54 peptide, central obesity, gender, and insulin dose adjusted by HbA1c were significant for predicting log (leptin) and all these variables had a positive linear relationship with log (leptin). Model 6 and 7 had similar predictors. The two models had similar findings with the univariate regressions (Table 17). Comparing the AIC, BIC, R 2 and adjusted R 2 of the two models, model 6 and 7 had similar AIC, R 2 and adjusted R 2 except BIC. The smaller number of observations might be a reason for the larger BIC of model 7. However, the two models showed similar significant predictors for predicting log (leptin) Final models of log (adiponectin/leptin) Table 20. Multivariate linear regression models of log (adiponectin/leptin) Model Predictors Beta coefficient DF p- value(overall/ individual) Number of predicto rs Number of observatio ns Root MSE R2 Adjuste d R2 AIC BIC < BMI z-score Insulin dose adjusted by HbA1c Gender Age <.0001 <25% Waist % 25%-75% 0.07 >75% HbA1c Glucose C-peptide Waist/height Gender BMI z-score

55 Two models were identified for log (adiponectin/leptin) and shown in Table 20. The final two models indicated that number of positive antibody was not a significant predictor of log (adiponectin/leptin). In model 8, insulin dose adjusted by HbA1c, gender, age and waist percentile were significant and all these variables except waist percentile had a negative linear relationship with log (adiponectin/leptin). Individuals in the highest waist percentile group (>75%) had lower log (adiponectin/leptin) levels and those in the 25-75% group had higher log (adiponectin/leptin) levels compared to individuals in the lowest waist group (<25%). In model 9, HbA1c, glucose, C-peptide, waist/height and gender were significant for predicting log (adiponectin/leptin) and only glucose had a positive linear relationship with log (adiponectin/leptin). Comparing the AIC, BIC, R 2 and adjusted R 2 of the two models, model 8 had higher R 2, higher adjusted R 2, smaller MSE root and smaller AIC and BIC. This indicated that model 8 fit better. The two models showed similar significant independent variables for predicting log (adiponectin/leptin) Results of multivariate regression models The final multivariate regression models gave consistent results with the univariate regression results (Table 17). For log (adiponectin), waist percentile, number of positive antibodies and age were significant predictors. When number of positive antibodies or age increased, log (adiponectin) level decreased. Individuals in the highest waist percentile group (>75%) had lower log (adiponectin) levels and those in the 25-75% group had higher log (adiponectin) levels compared to individuals in the lowest waist group (<25%). For log (leptin), BMI z-score, C-peptide, gender, HbA1c, central obesity, gender, and insulin dose adjusted by 41

56 HbA1c were significant predictors. For log (adiponectin/leptin), the significant predictors were gender, age, waist percentile, HbA1c, glucose, C-peptide and waist/height ratio. 3.6 TEST OF REGRESSION MODEL ASSUMPTIONS Existence For each specific combination of the predictors, log (adiponectin), log (leptin) and log (adiponectin/leptin) had normal distributions, according to Figure 1 and Table Independence Log (adiponectin), log (leptin) and log (adiponectin/leptin) were randomly distributed Linearity The partial regression plot of residuals versus each predictor with adjustment for all other predictors indicated if any nonlinearity was present in the relationship between outcome variable and each predictor. For log (adiponectin), log (leptin) and log (adiponectin/leptin), those plots between continuous predictors and the outcomes did not indicate a clear departure from linearity. 42

57 Linearity testing for log (adiponectin) The partial regression plots generally showed a linear relationship between the continuous predictors and the log (adiponectin). However there was variability as indicated by a noticeable amount of scatter around the regression line. (Figure 2 to 6). Figure 2. Partial regression residual plot for model 1 Figure 3. Partial regression residual plot for model 2 43

58 Figure 4. Partial regression residual plot for model 3 Figure 5. Partial regression residual plot for model 4 44

59 Figure 6. Partial regression residual plot for model Linearity testing for log (leptin) The partial regression plots generally showed a linear relationship between the continuous predictors and the log (leptin). However there was variability as indicated by a noticeable amount of scatter around the regression line. (Figure 7 to 8). 45

60 Figure 7. Partial regression residual plot for model 6 Figure 8. Partial regression residual plot for model 7 46

61 Linearity testing for log (adiponectin/leptin) The partial regression plots generally showed a linear relationship between the continuous predictors and the log (adiponectin/leptin). However there was variability as indicated by a noticeable amount of scatter around the regression line. (Figure 9 to 10). Figure 9. Partial regression residual plot for model 8 Figure 10. Partial regression residual plot for model 9 47

62 3.6.4 Homoscedasticity The homoscedasticity could be tested by looking if the variance of outcome variables were the same for any fixed combination of predictors. The two dimensional plots could help to test the homoscedaticity problem. There were two plots corresponding to each model. The first was plot between the residuals and the predicted outcomes. If the pattern was randomly distributed, the homoscedasticity assumption was fulfilled. The second plot was between the outcomes and the predicted outcomes. If a linear relationship was displayed, then the homoscedasticity assumption was fulfilled. From the following plots (Figure 12, Figure 14, Figure 16, Figure 18, Figure 20, Figure 22, Figure 24, Figure 26, Figure 28) of all nine models, the residuals were all randomly distributed versus the predicted outcomes. The positive linear relationships between the outcomes and the predicted outcomes were clearly displayed from the scatter plots of outcomes versus predicted outcomes of model 1 to 9(Figure 11, Figure 13, Figure 15, Figure 17, Figure 19, Figure 21, Figure 23, Figure 25, Figure 27). Figure 11. Model 1: Scatter plot of standardized residuals versus fitted points 48

63 Figure 12. Model 1: Scatter plot of outcome variables versus fitted points Figure 13. Model 2: Scatter plot of standardized residuals versus fitted points Figure 14. Model 2: Scatter plot of outcome variables versus fitted points 49

64 Figure 15. Model 3: Scatter plot of standardized residuals versus fitted points Figure 16. Model 3: Scatter plot of outcome variables versus fitted points Figure 17. Model 4: Scatter plot of standardized residuals versus fitted points 50

65 Figure 18. Model 4: The scatter plot of outcome variables versus fitted points Figure 19. Model 5: Scatter plot of standardized residuals versus fitted points Figure 20. Model 5: Scatter plot of outcome variables versus fitted points 51

66 Figure 21. Model 6: Scatter plot of standardized residuals versus fitted points Figure 22. Model 6: Scatter plot of outcome variables versus fitted points Figure 23. Model 7: Scatter plot of standardized residuals versus fitted points 52

67 Figure 24.Model 7: Scatter plot of outcome variables versus fitted points Figure 25.Model 8: Scatter plot of standardized residuals versus fitted points Figure 26. Model 8: Scatter plot of outcome variables versus fitted points 53

68 Figure 27. Model 9: Scatter plot of standardized residuals versus fitted points Figure 28. Model 9: Scatter plot of outcome variables versus fitted points Normality For any fixed combination of predictors, the outcome variables had normal distributions. The normality could be tested by plotting the residuals. One dimensional histogram and the QQ plot could reveal the normality of the residuals and helped to test if the outcome variables had normal distributions. The histogram and QQ plots (Figure 29-37) of the residuals indicated that the residuals were normally distributed, comparing with the standard normal distribution. The p-values of 54

69 Goodness-of-Fit Tests for normal distribution by Kolmogorov-Smirnov tests were all greater than 0.25, the normality assumption of all nine models were satisfied. Figure 29. Histogram and QQ plot of standardized residual versus normal distributions for model 1 Figure 30. Histogram and QQ plot of standardized residual versus normal distributions for model 2 55

70 Figure 31. Histogram and QQ plot of standardized residual versus normal distributions for model 3 Figure 32. Histogram and QQ plot of standardized residual versus normal distributions for model 4 Figure 33. Histogram and QQ plot of standardized residual versus normal distributions for model 5 56

71 Figure 34. The histogram and QQ plot of standardized residual versus normal distributions based on model 6 Figure 35. Histogram and QQ plot of standardized residual versus normal distributions for model 7 Figure 36. Histogram and QQ plot of standardized residual versus normal distributions for model 8 57

72 Figure 37. Histogram and QQ plot of standardized residual versus normal distributions for model Leverage and influence points diagnostics Leverage points The leverage points were outliers in the predictor space, it could be clarified by using the criteria hi>2(k+1)/n, where k= number of predictors in the model, n = observation numbers. The following plots (Figure 38) showed the leverage points labeled with patient numbers for all nine models. For model 3 and 5, there were more than 10 leverage points. For the other models, the leverage points were no more than 5, these observations should be checked to detect if there were some unusual records or tests. 58